Photon-echo synchronization and quantum state transfer in short quantum links

Abstract

The short quantum link regime, where the photon travel time $τ$ is comparable to the emitter lifetime $1/γ$, is experimentally relevant but theoretically underexplored: existing few-mode descriptions lose validity as retardation and multimode effects become significant. Using a Delay Differential Equation (DDE) framework that admits exact analytical solutions from the single-mode cavity limit to the multimode waveguide continuum, we show that emitters coupled to a short link spontaneously lock into self-synchronized Rabi oscillations driven by coherent photon echoes, breaking the link's discrete time-displacement symmetry. The resulting spectral structure -- persistent quasi-dark states and vacuum Rabi splitting, including in the superstrong coupling regime -- enables efficient quantum state transfer (QST): benchmarking three protocols across the full $γτ$ parameter space, we find that STIRAP exploits the quasi-dark-state structure to achieve a quadratic infidelity floor $\mathcal{O}((γτ)^2)$, outperforming both SWAP (linear error $\mathcal{O}(γτ)$) and wavepacket engineering for $γτ\lesssim 1.44$, even in regimes where retardation cannot be neglected. These results establish photon-echo synchronization as an engineering resource for quantum state transfer, with DDE modeling providing the exact analytical predictions needed to design and optimize short-link experiments on current circuit-QED hardware.

Figures (9)

• Figure 1: Minimal quantum link. (a) Two qubits interact with a closed waveguide of length $L$, with possibly time-dependent coupling strengths $\gamma_{1,2}(t)$. (b) The waveguide supports an approximately linear dispersion $\omega_k=ck$, with a free spectral range $\delta\omega_\text{FSR}=\pi c/L=\pi/\tau$ determined by the time $\tau$ a photon takes to travel from one of the waveguide to another.
• Figure 2: Dynamics of an initially excited two-level emitter in a medium-size cable. (a) Population of the excited state of the emitter as a function of time. (b) Time derivative of the population of the excited state for the first six periods of time. The solid black line represents the analytical solution of the DDE, and the dashed blue line shows the result of a numerical simulation using a Wigner-Weisskopf ansatz. The parameters taken were $\gamma \tau = 0.1$ and $\Delta = 50 \delta\omega_\text{FSR}$.
• Figure 3: Dynamics of an initially excited two-level emitter in a medium-size cable for different values of $\gamma \tau$. The solid black lines correspond to the solution of the DDE, while the dashed lines represent the solution using a truncated Hamiltonian.
• Figure 4: Power spectrum of the photons in the cavity as a function of the qubit's energy $\Delta$. In both cases, the orange dotted line is the analytical result of the eigenvalues, while the heat map is the result of a numerical analysis using the analytical solution. (a) weak coupling regime with $\gamma \tau = 0.15$. (b) strong coupling regime with $\gamma \tau = 1.5$.
• Figure 5: QST dynamics and efficiency comparison across the cavity-to-retardation crossover. Minimum infidelity $1-F$ for the three QST protocols at their respective optimal transfer times $T$ as a function of $\gamma_0\tau$. Each protocol is evaluated at its own optimal $T$ (SWAP: $T\sim\pi/\sqrt{\gamma_0\tau}$; STIRAP: $T\sim9/\sqrt{\gamma_0\tau}$; CZKM: $T=9/\sqrt{\gamma_0\tau}$), so the comparison is at different absolute durations. The dashed line denotes the exact CZKM lower bound on the infidelity. Symbols represent renormalized WW simulation results (corrected for Lamb-shift at finite detuning) at $\Delta/\delta\omega_\text{FSR}=50$ (circle), $5$ (diamond), and $1$ (triangle).